Spanish media reported the fee would be around 15 million euros ($17.4 million).
Silva, 21, is considered to be one of Brazil's most promising young players and has represented his country at junior level.
"We have been in touch with Real Madrid since last year and we're now discussing a fee," Cruzeiro's director of football Valdir Barbosa told ESPN radio. "I think a deal should be complete by the start of next week."
Silva was released from training on Thursday to have a medical with the European champions, said reports.
His arrival could prompt the departure of Asier Illarramendi who is believed to be a target for Real's struggling La Liga rivals Athletic Bilbao.
($1 = 0.8608 euros)
(Reporting by Iain Rogers in Madrid and Andrew Downie in Sao Paulo, editing by Tony Jimenez)
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